Furthermore, if the value of the useful work is greater than the cost of wasted work, then the mining incentives would be dominated by the value of the useful work. Though I haven't thoroughly examined the result of this scenario, I suspect that it could also result in the failure of the protocol.
Any kind of "work" is going to have value.
With bitcoin's current incentive structure, a miner will perform PoW "for" bitcoin users, who in turn "pay" the miner in the form of the block subsidy. Miners will also confirm transactions for a specific subset of bitcoin users, who will also pay the miners in the form of transaction fees. ASICs used to mine bitcoin cannot be used to perform any other
valuable work that comes close to the value of performing PoW "for" bitcoin users (performing PoW with the hopes of finding a block), so if the miners as a whole were to do something detrimental such as routinely accept double-spend transactions in a block race, or perform a 51% attack against bitcoin, the value of the ASIC mining equipment will decline to nearly zero.
On the other hand, if a bitcoin miner were performing some kind of PoW to try to find a bitcoin block (for the purpose of receiving the block reward -- that is the block subsidy plus tx fees), in addition to performing additional "useful" work that is valuable to someone else, the miners will not have the same incentives to not attack bitcoin. If the value of the other work is far in excess of the value of the EV of the block reward, a miner may seek short-term profit in attacking bitcoin, or they may not even bother with mining bitcoin. If the miners were to attack bitcoin, the value of their equipment would not drop to near zero because revenue could still be obtained via performing the other "useful" work.
There is a potential way to resolve these flaws. If the useful work is a public good, then the miners would gain no value from it and there would be no incentive to increase mining capacity based on the value of the useful work. The problem is that it is not clear that a true public good can exist. The protocol would be a failure if miners discovered a way to exploit the value of the useful work that was originally thought to be a public good.
A public good is typically paid for by the government. If a government cannot obtain a public good via its own resources (for example, a country that does not have the manufacturing capacity to produce tanks), will seek to exchange its abundant resources (which may include money) for resources that will produce said public good. Further, if a government cannot provide a public good period, its wealthy citizens will seek to use their own resources to obtain said public good amongst themselves (for example, a neighborhood building a security wall and hiring private security when the local government cannot handle crime). So there will always be someone willing to pay for a public good, and if you can provide a public good, you should expect to receive compensation for providing this service.